State estimation based on stochastic polynomials and variational approximation

Rafael Holdorf Lopez¹; José Eduardo Souza de Cursi²; André Carlon¹

¹ Universidade Federal de Santa Catarina; ² Institut National des Sciences Appliquées de Rouen

doi:10.20906/CPS/USM-2016-0005

Resumo

In many engineering applications, it becomes necessary to estimate the internal state of a system for the purposes of monitoring its health and/or achieving control of its dynamic behavior. Normally, the state estimation must be accomplished based on a few (sometimes indirect) measures of its state variables. These measures are in most cases affected by some level of noise/uncertainty. Typical methods to deal with such a problem are, for instance, least squares and/or Bayesian based approaches. In order to offer a different approach for the solution of state estimation problems, in this paper, we proposed a new method based on uncertainty quantification, more precisely, the representation of random variables using stochastic polynomials. The main idea of the proposed approach is to expand the state variables in terms of the random variables which represent the noise/uncertainty of the system, and then, apply a variational approximation to the iterative steps of a given numerical solution method (e.g. Euler, Runge-Kutta). The proposed approach is first presented to discrete systems, and then, it is extended to nonlinear continuous systems discretized by numerical methods. Three examples are analyzed in order to demonstrate the effectiveness of the proposed method, including the estimation of the state variables of the Hodgkin and Huxley's model. The results of this analysis show that the proposed estate estimation method correctly estimates the values of the state variables of dynamical systems subjected to noise/uncertainty.

Palavras-chave: State Estimation; Uncertainty Quantification; Stochastic Polynomials